On the containment problem for fat points

Abstract

Given an ideal $I$, the containment problem is concerned about finding the values $m$ and $n$ such that the $m$-th symbolic power of $I$ is contained in its $n$-th ordinary power. In this paper we consider this problem focusing on two classes of ideals of fat points. In particular we study the ideal of $n$ points on a line in ${\mathbb{P}}^N$ and the ideal of three nonlinear points in ${\mathbb{P}}^N$ for $N\ge 2$.